May  2011, 30(2): 547-558. doi: 10.3934/dcds.2011.30.547

Decay estimation for positive solutions of a $\gamma$-Laplace equation

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097

2. 

Department of Applied Mathematics, University of Colorado at Boulder

3. 

Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309

Received  June 2010 Published  February 2011

In this paper, we study the properties of the positive solutions of a $\gamma$-Laplace equation in $R^n$

-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$, $p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth function bounded by two positive constants. First, the positive solution $u$ of the $\gamma$-Laplace equation above satisfies an integral equation involving a Wolff potential. Based on this, we estimate the decay rate of the positive solutions of the $\gamma$-Laplace equation at infinity. A new method is introduced to fully explore the integrability result established recently by Ma, Chen and Li on Wolff type integral equations to derive the decay estimate.

Citation: Yutian Lei, Congming Li, Chao Ma. Decay estimation for positive solutions of a $\gamma$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 547-558. doi: 10.3934/dcds.2011.30.547
References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304.

[2]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[3]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: 10.2307/2951844.

[4]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5.

[5]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, preprint, (2009).

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. in Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.

[8]

C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities,, Potential Analysis, 16 (2002), 347. doi: 10.1023/A:1014845728367.

[9]

A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91.

[10]

S. Ding, On some imbedding theorems,, Sci. Sinica, 21 (1978), 287.

[11]

L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,'', Cambridge Unversity Press, (2000). doi: 10.1017/CBO9780511569203.

[12]

M. Franca, Classification of positive solutions of p-Laplace equation with a growth term,, Archivum Mathematicum, 40 (2004), 415.

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, 7a (1981).

[14]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.

[15]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.

[16]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447.

[17]

T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591.

[18]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793.

[19]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.

[20]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453.

[21]

C. Li and L. Ma, Uniqueness of positive bound states to Schrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.

[22]

Y. Li, Remark on some conformally invariant integral equations: the method of moving planes,, Journal of European Mathematical Society, 6 (2004), 153. doi: 10.4171/JEMS/6.

[23]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032.

[24]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020.

[25]

J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513. doi: 10.1007/s00229-003-0358-4.

[26]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859.

[27]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: 10.1007/BF00250468.

[28]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79. doi: 10.1007/BF02392645.

[29]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.

show all references

References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304.

[2]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[3]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: 10.2307/2951844.

[4]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5.

[5]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, preprint, (2009).

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. in Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.

[8]

C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities,, Potential Analysis, 16 (2002), 347. doi: 10.1023/A:1014845728367.

[9]

A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91.

[10]

S. Ding, On some imbedding theorems,, Sci. Sinica, 21 (1978), 287.

[11]

L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,'', Cambridge Unversity Press, (2000). doi: 10.1017/CBO9780511569203.

[12]

M. Franca, Classification of positive solutions of p-Laplace equation with a growth term,, Archivum Mathematicum, 40 (2004), 415.

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, 7a (1981).

[14]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.

[15]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.

[16]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447.

[17]

T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591.

[18]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793.

[19]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.

[20]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453.

[21]

C. Li and L. Ma, Uniqueness of positive bound states to Schrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.

[22]

Y. Li, Remark on some conformally invariant integral equations: the method of moving planes,, Journal of European Mathematical Society, 6 (2004), 153. doi: 10.4171/JEMS/6.

[23]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032.

[24]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020.

[25]

J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513. doi: 10.1007/s00229-003-0358-4.

[26]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859.

[27]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: 10.1007/BF00250468.

[28]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79. doi: 10.1007/BF02392645.

[29]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.

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